Square Root of 225
2026-02-28 01:09 Diff
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Last updated on August 5, 2025

The square root of 225 is a value “y” such that when “y” is multiplied by itself → y ⤫ y, the result is 225. The number 225 has a unique non-negative square root, called the principal square root.

What Is the Square root of 225?

The square root of 225 is ±15, where 15 is the positive solution of the equation x2 = 225. Finding the square root is just the inverse of squaring a number and hence, squaring 15 will result in 225. The square root of 225 is written as √225 in radical form, where the ‘√’  sign is called the  “radical” sign. In exponential form, it is written as (225)1/2 
 

Finding the Square Root of 225

We can find the square root of 225 through various methods. They are:
i) Prime factorization method
ii) Long division method
iii) Repeated subtraction method.
 

Square Root of 225 By Prime Factorization Method

The prime factorization of 225 can be found by dividing the number by prime numbers and continuing to divide the quotients until they can’t be separated anymore.

After factorizing 225, make pairs out of the factors to get the square root.

So, Prime factorization of 225 = 3 × 5 × 3 × 5

Square root of 225 = √[3 × 3 ×5 × 5] = 3 × 5= 15

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Square Root of 225 By Long Division Method

This method is used for obtaining the square root for non-perfect squares, mainly.

It usually involves the division of the dividend by the divisor, getting a quotient and a remainder too sometimes.

Follow the steps to calculate the square root of 225:

Step 1: Write the number 225 and draw a bar above the pair of digits from right to left.

Step 2: Now, find the greatest number whose square is less than or equal to 2. Here, it is 1 because 12=1 < 2.

Step 3: now divide 225 by 1 (the number we got from Step 2) such that we get 1 as a quotient, and we get a remainder.  

Double the divisor 1, we get 2, and then the largest possible number A1=5 is chosen such that when 5 is written beside the new divisor 2, a 2-digit number is formed →25, and multiplying 5 with 25 gives 125, which when subtracted from 125, gives 0.

Repeat this process until you reach the remainder of 0. 

Step 4: The quotient obtained is the square root of 225. In this case, it is 15.

Square Root of 225 By Subtraction Method

We know that the sum of the first n odd numbers is n2. We will use this fact to find square roots through the repeated subtraction method.

Furthermore, we just have to subtract consecutive odd numbers from the given number, starting from 1. The square root of the given number will be a count of the number of steps required to obtain 0.

Here are the steps:

Step 1: Take the number 225 and then subtract the first odd number from it. Here, in this case, it is 225-1=224.

Step 2: we have to subtract the next odd number from the obtained number until it comes zero as a result. Now take the obtained number (from Step 1), i.e., 224, and again subtract the next odd number after 1, from 3, i.e.,224-3=221. Like this, we have to proceed further.

Step 3: Now we have to count the number of subtraction steps it takes to yield 0 finally. Here, in this case, it takes 15 steps. So, the square root is equal to the count, i.e., the square root of 225 is ±15.


Common Mistakes and How to Avoid Them in the Square Root of 225

When we find the square root of 225, we often make some key mistakes, especially when we solve problems related to that. So, let’s see some common mistakes and their solutions
 

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Problem 1

Find the radius of a circle whose area is 225π² cm².

Okay, lets begin

Given, the area of the circle = 225π cm2
Now, area = πr2 (r is the radius of the circle)
So,  πr2 = 225π cm2
We get, r2 = 225 cm2
r = √225 cm


Putting the value of √225 in the above equation, 
We get, r = ±15 cm
Here we will consider the positive value of 15.


Therefore, the radius of the circle is 15 cm.

Answer: 15 cm.
 

Explanation

We know that, area of a circle = πr2 (r is the radius of the circle). According to this equation, we are getting the value of “r” as 15 cm by finding the value of the square root of 15
 

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Problem 2

Find the length of a side of a square whose area is 225 cm²

Okay, lets begin

Given, the area = 225 cm2
We know that, (side of a square)2 = area of square
Or,  (side of a square)2 = 225
Or,  (side of a square)= √225
Or, the side of a square = ± 15.


But, the length of a square is a positive quantity only, so, the length of the side is15 cm.
Answer: 15 cm
 

Explanation

We know that, (side of a square)2 = area of square. Here, we are given with the area of the square, so, we can easily find out its square root because its Square root is the measure of the side of the square 

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Problem 3

Simplify the expression: √225 X √225, √225+√225

Okay, lets begin

√225x√225 =  √(15×15) x √(15×15)
=  15×15
=  225

                   √225+√225 = √(15×15)  + √(15×15) 
 = 15 + 15
 = 30
Answer: 225, 30

Explanation

In the first expression, we multiplied the value of the square root of 225 with itself. In the second expression, we added the value of the square root of 225 with itself.

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Problem 4

If y=√225, find y²

Okay, lets begin

Firstly, y=√225= 15

 Now, squaring y, we get, 
 y2=152=225
 or, y2=225
Answer : 225
 

Explanation

Squaring “y” which is same as squaring the value of √225 resulted to 225
 

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Problem 5

Calculate (√225/5 + √225/3)

Okay, lets begin

√225/5 + √225/3= 15/5 + 15/3
 = 3 + 5
 = 8
Answer: 8
 

Explanation

From the given expression, we first found the value of square root of 225 then solved by simple divisions and then simple addition

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FAQs on 225 Square Root

1.Is 225 a rational number?

2.Is 225 a perfect square or a non-perfect square?

 225 is a perfect square, since 225 =15 2
 

3.Is the square root of 225 a rational or irrational number?

The square root of 225 is ±15. So, 15 is a rational number since it can be obtained by dividing two integers and can be written in the form 15/1           

4.What is the cube root of 225?

 6.082201… is the cube root of 225.
 

5.How does the square root of 225 compare to the square root of other numbers?

 √100 = ±10, √25 = ±5, √16 =± 4
The square root of a perfect square is always an integer
 

6.What is the square root of 225 in decimal form?

 The square root of 225 in decimal form is 15.0
 

7.How to solve √224?

We can find the square root of 224 through different methods like Long Division method, Approximation Method or Prime Factorization Method. The value of √224 is 14.96666....

Important Glossaries for Square Root of 225

  • Exponential form

An algebraic expression that includes an exponent. It is a way of expressing the numbers raised to some power of their factors. It includes continuous multiplication involving base and exponent.
Ex: 2 ⤬ 2 ⤬ 2 ⤬ 2 = 16
Or, 2 4 = 16, where 2 is the base, 4 is the exponent

  • Factorization   

Expressing the given expression as a product of its factors
Ex: 48=2 ⤬ 2 ⤬ 2 ⤬ 2 ⤬ 3

  • Prime Numbers 

Numbers which are greater than 1, having only 2 factors as →1 and Itself. Ex: 1,3,5,7,….

  •  Rational numbers and Irrational numbers

The Number which can be expressed as p/q, where p and q are integers and q not equal to 0 are called Rational numbers. Numbers which cannot be expressed as p/q, where p and q are integers and q not equal to 0 are called Irrational numbers. 

  •  Perfect and non-perfect square numbers

Perfect square numbers are those numbers whose square roots do not include decimal places. Ex: 4,9,25 Non-perfect square numbers are those numbers whose square roots comprise decimal places. Ex :3, 8, 24
 

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Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

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: He loves to play the quiz with kids through algebra to make kids love it.